Existence and nonexistence results for anisotropic quasilinear elliptic equations

Existence and nonexistence results for anisotropic quasilinear elliptic equations

Ilaria Fragalà

, Filippo Gazzola

, Bernd Kawohl

aDipartimento di Matematica del Politecnico, piazza Leonardo da Vinci 32, 20133 Milano, Italy bMathematisches Institut, Universität Köln, 50923 Köln, Germany

Received 13 October 2002 Available online 27 February 2004

Abstract

We consider a new class of quasilinear elliptic equations with a power-like reaction term: the differential operator weights partial derivatives with different powers, so that the underlying functional-analytic framework involves anisotropic Sobolev spaces. Critical exponents for embeddings of these spaces intoL^q have two distinct expressions according to whether the anisotropy is “concentrated” or “spread out”. Existence results in the subcritical case are influenced by this phenomenon. On the other hand, nonexistence results are obtained in the at least critical case in domains with a geometric property which modifies the standard notion of starshapedness.

2004 Elsevier SAS. All rights reserved.

Résumé

Nous considérons une nouvelle classe d’équations elliptiques quasilinéaires avec un terme de réaction de type puissance : les dérivées partielles ont des puissances différentes dans l’opérateur différentiel, de façon que l’espace fonctionnel naturel devient un espace de Sobolev anisotrope. Les exposants critiques pour les injections de ces espaces dansL^q ont des expressions différentes qui dépendent de la “concentration” de l’anisotropie. Nos résultats d’existence dans le cas sous-critique sont influencés par ce phenomène. D’autre part, nos résultats de non existence dans le cas critique et sur-critique sont obtenus dans des domaines ayant une propriété qui modifie la notion usuelle d’ensemble étoilé.

2004 Elsevier SAS. All rights reserved.

MSC: 35J70; 46E35; 35B65

Keywords: Anisotropic Sobolev spaces; Critical exponents; Minimax methods; Pohožaev identity

1. Introduction

We are interested in existence and nonexistence results for the following anisotropic quasilinear elliptic problem







−_n

i=1∂_i(|∂_iu|^mi−2∂_iu)=λu^p−1 inΩ,

u0 inΩ,

u=0 on∂Ω,

(1)

E-mail addresses: [email protected] (I. Fragalà), [email protected] (F. Gazzola), [email protected] (B. Kawohl).

0294-1449/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2003.12.001

whereΩ⊂Rⁿ (n2) is a smooth bounded domain,mi >1 for alli,λ >0 andp >1. Note that ifmi=2 for alli, then (1) reduces to the well-known semilinear equation−u=λu^p−1.

There is by now a large number of papers and an increasing interest about anisotropic problems. With no hope of being complete, let us mention some pioneering works on anisotropic Sobolev spaces [14,19,24,25,27] and some more recent regularity results for minimizers of anisotropic functionals [1,5,10,17,18,28].

Historically, the study of the semilinear problem−u=f (x, u)started by settling the background of a rigorous functional-analytic framework (Sobolev spaces) and by establishing the existence of solutions in a variational way, that is, minimizing suitable functionals. But then, the following step was to find solutions by means of minimax methods such as Birkhoff theory, Ljusternik–Schnirelmann category, mountain-pass and linking theorems. As far as we are aware, minimax methods have not yet been used for problems like (1), so the present paper is a first contribution in this direction.

A further motivation for the study of (1) is given by the necessity of an explanation of the link between quasilinear elliptic equations and embedding inequalities. It is known that for quasilinear elliptic equations involving them-Laplace operator_m(m >1), power-like reaction terms exhibit several critical exponents, see [9]

and references therein. More precisely, critical exponents of suitable embedding inequalities are also the borderline between existence and nonexistence results for solutions of such equations. Therefore, one may wonder if these results may be obtained only using functional analysis, without exploiting the typical features of elliptic operators such as regularity theory, maximum principles, homogeneous eigenvalue problems. And the elliptic operator in (1) precisely fails to possess these properties.

Our starting point is the observation that embedding theorems for anisotropic Sobolev spaces occur below a critical exponent which has a different value if the anisotropy is spread out or concentrated. More precisely, let m=(m₁, . . . , m_n)and denote byW₀^1,m(Ω)the closure ofC_c^∞(Ω)with respect to the norm

u1,m= n i=1

∂_iumi.

When the exponentsmi are not “too far apart”, the critical exponentm^∗for the embeddingW₀^1,m(Ω)⊂L^q(Ω)is just the usual critical exponent corresponding to the harmonic mean of themi; on the other hand, if themi are “too much spread out” it coincides with the maximumm₊of themi. Therefore the effective critical exponent is in fact the maximum of these two values,m_∞=max{m^∗, m₊}, see Theorem 1 below. Existence results for (1) are quite different in the two mentioned situations.

However, before wondering about existence results, due to the lack of a satisfactory regularity theory, one must be careful in describing what is meant by a solution of (1). In the next section (Definition 1), we introduce three different kinds of solutions, weak, mild, and strong, according to their summability. In Theorem 2 we prove that, in the subcritical case, weak solutions of (1) are actually strong, namely they are summable at any power. In order to prove this fact, due to the anisotropy of the differential operator, we need several essential modifications of the method developed by Brezis and Kato in [4].

Once the different kinds of solutions are clarified, we may turn to existence results. In Theorems 3 and 4 we apply respectively constrained minimization methods and the mountain-pass Theorem in order to prove the existence of strong solutions of (1) in the “compact” case. It turns out that also the application of these by now standard tools is not straightforward. First of all, the “kinetic functional” (which coincides with the Dirichlet integral whenm_i ≡2) is not homogeneous and rescaling is not allowed. Therefore, the minimization method merely enables us to find someλfor which (1) admits nontrivial solutions. Moreover, it is not clear which exponents p yield a resonance situation, i.e. eigenvalue problems, see Problem 2 in Section 8.3. On the other hand, the application of the mountain-pass Theorem requires further restrictions on the exponentsm_i, see (5) below and the remarks in Section 8.1.

In order to prove nonexistence results for at least critical growth problems, the most common tool is the celebrated Pohožaev identity [21,22]. However, even its weaker formulations require solutions of class C¹(Ω)

in order to have well-defined boundary terms, see [7,8,12]. And it seems a challenging problem to obtain such regularity for weak solutions of (1), see [10]. To overcome this difficulty we introduce a sequence of “doubly approximating” problems inspired by a nice idea of Otani [20]. This procedure turns out to be quite delicate, due to the anisotropy of the operator. Indeed, we need to prove a strong regularity result for the approximating problems, see Theorem 5. When the approximation procedure is over, we are able to prove our main nonexistence result, see Theorem 6. It states that, in the at least critical case, (1) admits no mild solutions other thanu≡0. This result requires two assumptions of different kind. First, the domainΩmust have a new geometrical feature, which modifies the classical notion of starshapedness according to the anisotropy of the operator; we call this property α-starshapedness and we feel that it sheds some light on the interplay between the structure of the differential operator and the geometry of the domain. Second, the exponentsm_imust be sufficiently concentrated: this technical assumption, which might probably be relaxed (see Problem 3), guarantees the regularity of solutions to the coercive approximating problem.

The precise statements of the results are given in Section 2, and their proofs are postponed to the subsequent sections. Finally in Section 8, we collect some further remarks and we address some related open problems.

2. Results

2.1. Functional setting and summability of solutions

Throughout the paper we assume thatΩ is an open bounded domain with (at least) Lipschitz boundary∂Ω, and we denote by( , )the Euclidean scalar product on Rⁿ. We also always assume, without recalling it at each statement, that the exponentspandm_iappearing in (1) satisfy the conditions

p >1, mi>1 ∀i=1, . . . , n,

n i=1

1

m_i >1. (2)

The last condition in (2) ensures that the anisotropic Sobolev space W₀^1,m(Ω) embeds into some Lebesgue spacesL^q(Ω); if it is violated, one has embeddings into Orlicz’s or Hölder’s spaces. Embeddings of the kind W₀^1,m(Ω)⊂L^q(Ω) are in fact a fundamental tool to study the existence of solutions for the boundary value problem (1). Let us set

m^∗= n

_n

i=1 1

mi −1, m₊:=max{m1, . . . , mn}, m_∞=max{m₊, m^∗}. (3) Note thatm^∗is well-defined thanks to (2), and that it coincides with the usual critical exponentm^∗:=nm/(n − m) for the harmonic meanmof themi. Note also that it may well happen thatm₊> m^∗(this occurs for instance if n=4,m1=m2=m3=2,m₊=m4=100), thus it is meaningful to define the maximal exponentm_∞. Actually, m_∞ turns out to be the “true” critical exponent. In Section 1 we prove the following result, which we could not find in the literature.

Theorem 1. LetΩ⊂Rⁿbe an open bounded domain with Lipschitz boundary. If (2) holds, then for allq∈ [1, m_∞] there is a continuous embeddingW₀^1,m(Ω)⊂L^q(Ω). Forq < m_∞, the embedding is compact.

Remark 1. Theorem 1 is no longer true if the zero trace condition on the boundary is removed. More precisely, denote by W^1,m(Ω) the closure of the restrictions to Ω of functions in C_c^∞(Rⁿ) with respect to the norm · 1,m+ · 1. Then, even for smooth domainsΩ, in order to have the embeddingW^1,m(Ω)⊂L^m∗(Ω)some geometric restrictions onΩ are needed, see e.g. [14,19,24,25].

We are now going to characterize three different kinds of solutions to the boundary value problem (1). To this end, we also need to consider the smallest exponent

m₋:=min{m₁, . . . , mn},

and, for givenq∈ [1,+∞], we denote byq :=q/(q−1)its conjugate exponent.

Definition 1. We say thatu∈W₀^1,m(Ω)∩L^(p−1)m∞(Ω)is a weak solution of (1) ifu0 a.e. inΩand n

i=1

Ω

|∂iu|^mi−2∂iu ∂iv=λ

Ω

u^p−1v ∀v∈W₀^1,m(Ω). (4)

If in additionu∈L^(p−1)m−(Ω), we say thatuis a mild solution. Finally, ifu∈L^∞(Ω)we say thatuis a strong solution.

Clearly, every strong solution is also a mild solution, and the latter is also a weak solution. In some cases, we may prove the converse implications:

Theorem 2. If one of the two following situations occurs (i) p < m_∞,

(ii) p=m_∞andm_∞> m₊,

then every weak solution of (1) is also a strong solution.

A proof of Theorem 2 is given in Section 4. For related results concerning local minimizers, we refer to [5, Theorem 2]. We believe that Theorem 2 holds under the mere assumptionpm_∞, see Problem 1 in Section 8.3.

In Section 8.2 we discuss an example which suggests the kind of solutions we should expect, according to the value ofp. We also stress that, in the semilinear case (i.e.m_i≡2), elliptic theory enables one to show that a strong solution of (1) in a smooth domain is a classical solution inC²(Ω), but for general mi this regularity seems out of reach.

2.2. Existence results

First of all, we remark that it is not clear whichp yields the so-called resonance for (1). Namely, is there somepwhich gives rise to a “generalized eigenvalue” problem? Obviously, ifm₊=m₋, the resonance problem corresponds top=m₋, see [3]. In the general case, we have

Theorem 3. Assume thatp < m_∞. Then, for anyγ >0 there existλ_γ>0 andu_γ∈W₀^1,m(Ω)such thatu_γp=γ andu_γ is a strong solution of (1) whenλ=λ_γ.

In other words, there exists a continuum of pairs(λ_γ, u_γ)∈(0,∞)×W₀^1,m(Ω)which solve (1), seen as an eigenvalue problem. We point out that Theorem 3 cannot be used to deduce the existence of a solution to problem (1) for a givenλ. In fact, unless all them_i are equal, rescaling methods fail due to the lack of homogeneity of the differential operator.

Then, to recover an existence result for fixedλ, we apply the mountain-pass Theorem [2]. In order to deal with a “superlinear” subcritical problem we need to assume that

m₊< m^∗. (5)

Note that ifmi=m₊forn−1 indicesi, then (5) is automatically fulfilled; in particular, it holds ifn=2.

Then we prove:

Theorem 4. Assume that the exponentsmisatisfy assumption (5) and letp∈(m₊, m^∗). Then, for allλ >0 problem (1) admits a nontrivial strong solution.

Due to assumption (5), this statement is probably not optimal, but it seems not clear at all which are the sharp assumptions that ensure both a mountain-pass geometric structure and the Palais-Smale condition for the involved functional, see Problem 2 in Section 8.3. In Section 8.1 we exhibit two examples where the assumptions of Theorem 4 are violated and the mountain-pass Theorem cannot be applied.

2.3. Regularity and nonexistence results

We now require that themi satisfy the additional assumption

m_i2 (6)

and the “not too far apart condition”

m₊<n+2

n m₋. (7)

Note that if (6) and (7) hold, we necessarily haven3 and (5), so thatm_∞=m^∗. In order to establish our main nonexistence result, we consider some approximating problems, which are coercive and uniformly elliptic, and we prove that they admit a unique and smooth solution.

Theorem 5. Assume that∂Ω∈C^2,γ, and that the exponentsm_i satisfy assumptions (6) and (7). Letp >1,λ >0 andf ∈C_c^∞(Ω). Then, for allε >0, the problem

−n

i=1∂i[(|∂iw|^mi−2+ε(1+ |Dw|²)^(m−−2)/2)∂iw] +λ|w|^p−2w=f inΩ,

w=0 on∂Ω

(8) admits a unique (classical) solutionw∈C²(Ω).

We finally turn to the at least critical casepm^∗. We prove nonexistence results in domains which haveC^2,γ boundary and satisfy the following geometrical condition.

Definition 2. Letα=(α1, . . . , αn)∈Rⁿ withαi >0 for alli. We say that a bounded smooth domainΩ⊂Rⁿis α-starshaped with respect to the origin if

n i=1

αixiνi0 on∂Ω, (9)

withν=(ν1, . . . , νn)denoting the outer normal to∂Ω. We say thatΩ is strictlyα-starshaped with respect to the origin if (9) holds with strict inequality. If these inequalities hold after replacingx_i byx_i−P_i, we say thatΩ is (strictly)α-starshaped with respect to the centerP =(P₁, . . . , P_n). IfΩ is (strictly)α-starshaped with respect to some of its points, we simply say thatΩis (strictly)α-starshaped.

Several remarks about this notion of “anisotropic starshapedness” are in order. We collect them in Section 2.5.

Since the solution in Theorem 5 is smooth up to the boundary, we may write the Pohožaev identity, see Proposition 1. Then, thanks to a suitable double passage to the limit, we prove:

Theorem 6. Assume that ∂Ω ∈C^2,γ, and that the exponents m_i satisfy assumptions (6) and (7). Let α= (α₁, . . . , α_n)with

αi=n 1 m_i − 1

m^∗

.

Assume that eitherp > m^∗ andΩ isα-starshaped, orp=m^∗ andΩ is strictly α-starshaped. Then, for every λ >0, the unique mild solution of(1)isu≡0.

Note that by (5) the α_i in Theorem 6 are all strictly positive. If m₊=m₋, then α_i =1 for all i, and α- starshapedness reduces to standard starshapedness.

2.4. Miscellaneous consequences

Thanks to Theorem 2, we can slightly improve Theorem 6 whenp=m^∗.

Corollary 1. Assume that ∂Ω ∈C^2,γ, and that the exponents mi satisfy assumptions (6) and (7). Let α= (α1, . . . , αn) be as in Theorem 6. Assume thatp=m^∗ andΩ is strictly α-starshaped. Then, for every λ >0, the unique weak solution of (1) isu≡0.

As already mentioned, existence results are strongly affected by the validity of condition (5); notice indeed that, as a consequence of Theorems 3 and 6, there holds

Corollary 2. Ifp=m^∗< m₊, then for any domainΩ there existsλ >0 such that(1)admits a nontrivial strong solution. Ifp=m^∗> m₊, and (6) and (7) hold, then there exist domainsΩ such that for everyλ >0 the unique weak solution of (1) isu≡0.

By analyzing the proof of Theorem 6, we realize that in some cases we may state a stronger result, which excludes also the existence of sign-changing solutions. Indeed we have:

Corollary 3. Assume that ∂Ω ∈C^2,γ, and that the exponents mi satisfy assumptions (6) and (7). Let α= (α1, . . . , αn)be as in Theorem 6. Assume that p > m^∗ andΩ isα-starshaped. Thenu≡0 is the unique mild solution to the problem

−_n

i=1∂i(|∂iu|^mi−2∂iu)=λ|u|^p−2u inΩ,

u=0 on∂Ω.

2.5. Aboutα-starshapedness

The notion ofα-starshapedness with respect to the centerP may be reformulated in a more geometric way as Tα(x−P ), ν

=(x−P , Tαν)0 on∂Ω, (10)

where T_α denotes the second order tensor _n

i=1α_ie_i ⊗e_i. In the following, by starshapedness we mean the classical notion, which corresponds to our definition when all the α_i coincide, namely when the tensor T_α is a positive multiple of the identity matrix. Some basic differences betweenα-starshapedness and (ordinary) starshapedness as well as the relationship between the two notions reveal themselves by looking at simple examples in dimensionn=2.

Fig. 1. Fig. 2.

Fig. 3. Fig. 4.

Example 1. The simplest example ofα-starshaped domain is a ballB: it is immediate that it isα-starshaped with respect to its centerOfor any choice ofα. Nevertheless, ifα₁=α₂, and we move the centerP ofα-starshapedness away from the centerO of the ball,B may result notα-starshaped with respect toP (recalling (10), see Fig. 1).

This shows that, as it happens for starshapedness, the notion ofα-starshapedness is sensitive to the choice of the center. But, in contrast to starshapedness, even a convex domain may be notα-starshaped with respect to some of its points.

Example 2. Consider an ellipseEwith equationax²+y²<1 (a >0). One can check thatEisα-starshaped with respect toO=(0,0)for everyα. Now rotateEclockwise by an angleπ/4: the rotated domainE may be no longer α-starshaped with respect toO(recalling (10), see Fig. 2). Thus, in contrast to starshapedness,α-starshapedness is not invariant under rotations.

Example 3. For fixed α with α₁ =α₂, it may happen that a domain is starshaped with respect to some center, but not α-starshaped with respect to any center. For instance, consider the set M represented in Fig. 3.

Clearly, it is starshaped with respect to O. However, take α withα₁α₂, and let P ∈M. To make the sum α1(x1−P1)ν1+α2(x2−P2)ν2positive on∂M, the pointP should belong to both shadowed subsets ofM. The intersection between such regions is empty.

Example 4. Again for fixedαwithα₁=α₂, the converse situation with respect to the previous example may occur, that is, it may happen that a domain isα-starshaped with respect to some center, but not starshaped with respect to any center. Consider for instance the setN represented in Fig. 4. Since the productx₁ν₁ remains positive on the whole boundary ofN, choosingαwithα₁α₂ the conditionα₁x₁ν₁+α₂x₂ν₂>0 is satisfied on∂N, soN is strictlyα-starshaped with respect toO. On the other hand,N cannot be starshaped with respect to anyP, because such aP should belong to the intersection of the two disjoint shadowed subsets ofN.

3. Proof of Theorem 1

The continuity of the embeddingW₀^1,m(Ω)⊂L^m+(Ω)relies on a well-known Poincaré-type inequality. More precisely, denoting by{e₁, . . . , e_n}the canonical basis of Rⁿ, assume thatΩhas widtha >0 in the direction ofe_i, namely sup_x,y∈Ω (x−y, e_i)=a. We claim that, for everyq1, we have

uqaq

2 ∂_iuq ∀u∈C_c¹(Ω). (11)

We prove (11) in the caseq >1, the caseq =1 being simpler. Assume without loss of generality thatΩ⊂ {x∈ Rⁿ;0< xi< a}, and, for allx∈Rⁿ, setx=(xi, x)in order to emphasize itsi-th component. Letu∈C_c¹(Ω)and letv(x)=u(x)∂iu(x). We consideru(andv) as defined on the whole Rⁿ, set to 0 outside spt(u). Denote byv⁺ (respectivelyv⁻) the positive part (respectively negative part) ofv. Then, we have

0=|u(a, x)|^q− |u(0, x)|^q q

= a 0

u(t, x)^q−2v(t, x) dt

= a 0

u(t, x)^q−2v⁺(t, x) dt+ a 0

u(t, x)^q−2v⁻(t, x) dt and

a 0

u(t, x)^q−2v⁺(t, x ) dt− a 0

u(t, x)^q−2v⁻(t, x) dt= a 0

u(t, x)^q−2v(t, x)dt which show that

a 0

u(t, x)^q−2v⁺(t, x ) dt=1 2 a 0

u(t, x)^q−2v(t, x)dt.

Therefore, we also have u(x_i, x)^q=q

xi

0

u(t, x)^q−2v(t, x) dtq

xi

0

u(t, x)^q−2v⁺(t, x) dt

q

a 0

u(t, x)^q−2v⁺(t, x) dt=q 2

a 0

u(t, x)^q−1∂iu(t, x)dt.

Hence, by integrating first with respect toxi∈(0, a)and then with respect tox ∈Rⁿ⁻¹we obtain u^qqaq

2 u^qq⁻¹∂_iuq

via Hölder’s inequality and (11) follows. Hence, by density, the embeddingW₀^1,m(Ω)⊂L^m+(Ω)is continuous. On the other hand, for the continuity of the embeddingW₀^1,m(Ω)⊂L^m∗(Ω), we refer to [27, Teorema 1.2] and [29, Corollary 2]. Thus, the embeddingW₀^1,m(Ω)⊂L^m∞(Ω)is also continuous.

In order to show the compactness of the embedding W₀^1,m(Ω)⊂L^q(Ω) for q < m_∞, we combine the continuous embeddingW₀^1,m(Ω)⊂W₀^1,m−(Ω)with the compact embeddingW₀^1,m−(Ω)⊂L¹(Ω)to deduce the compact embeddingW₀^1,m(Ω)⊂L¹(Ω). Then we conclude by interpolation betweenL¹(Ω)andL^m∞(Ω). 2

4. Proof of Theorem 2

We first show that any weak solution of (1) belongs toL^q(Ω)for allq∈ [1,∞). We have two different proofs under assumptions (i) and (ii), and we begin with

(ii) The casem₊< m^∗=p.

Letu be a weak solution to (1). The assertion thatu belongs toL^q(Ω)for allq <∞ may be equivalently reformulated as

u∈L^(a+1)m∗(Ω) for alla >0. (12)

By Theorem 1, to have (12) it is enough to show thatu^a+1∈W₀^1,m(Ω), which is in turn equivalent to

L→+∞lim n i=1

Ω

∂i

u·min[u^a, L]^mi1/mi

<+∞. (13)

In any case (i.e. if the l.h.s. of (13) is bounded or unbounded), asL→ ∞, up to a subsequence, there exists at least one indexj such that

n i=1

Ω

∂_i

u·min[u^a, L]^mi1/m_i

C

Ω

∂_j

u·min[u^a, L]^mj1/m_j

, (14)

where C denotes some positive constant independent of L. Fix such an index j, and, for every L >0, set ϕL:=u·min[u^amj, L^mj] ∈W₀^1,m(Ω). Note that

|∂_iu|^mi−2∂_iu∂_iϕ_Lmin[u^amj, L^mj]|∂_iu|^mi for a.e.x∈Ω, ∀i=1, . . . , n, (15) and ∂_i

u·min[u^a, L]^mi(a+1)^mimin[u^ami, L^mi]|∂_iu|^mi for a.e.x∈Ω,∀i=1, . . . , n. (16) Test (1) withϕL, integrate by parts and use (15), Hölder’s inequality and Theorem 1 to obtain (for anyk >0)

n i=1

Ω

min[u^amj, L^mj] · |∂iu|^mi

n

i=1

Ω

|∂_iu|^mi−2∂_iu ∂_iϕ_L=λ

Ω

u^m∗·min[u^amj, L^mj]

=C_k+λ

uk

u^m∗−mju^mj·min[u^amj, L^mj]

C_k+λ

uk

u^m∗

(m^∗−m_j)/m^∗

·

uk

u^mj ·min[u^amj, L^mj]m^∗/m_jm_j/m^∗

Ck+εk

Ω

u·min[u^a, L]m^∗mj/m^∗

C_k+ε_{k n}

i=1

Ω

∂_i

u·min[u^a, L]^mi_1/mim_j

,

where C_k → ∞ and ε_k →0 as k→ ∞ and they may denote different constants from line to line (with C_k independent ofLprovided one takes L > k^a). We also stress that in applying the Hölder’s inequality, we have used the assumptionm₊< m^∗). From the last inequality and from (16) we infer (forL > k^a)

Ω

∂j

u·min[u^a, L]^mjCk+εk

_n

i=1

Ω

∂i

u·min[u^a, L]^mi1/mim_j

. (17)

Inserting (14) into (17), we get

Ω

∂_j

u·min[u^a, L]^mjC_k+ε_k

Ω

∂_j

u·min[u^a, L]^mj.

Choosingk sufficiently large (i.e. ε_k sufficiently small), this shows that the r.h.s. of (14) remains bounded as L→ +∞and (13) follows.

(i) The casep < m_∞.

Letube a weak solution to (1). We claim that, if the implication

u∈L^am++p(Ω) ⇒ u∈L^(a+1)m∞(Ω) (18)

holds for alla >0, thenu∈L^q(Ω)for allq <∞. Indeed, define the sequence{ak}by setting a₀=m_∞−p

m₊ , a_k+1=m_∞

m₊a_k+m_∞−p m₊ .

Sinceak→ +∞(thanks to the assumptionp < m_∞), applying (18) witha=ak, we deduce thatu∈L^q(Ω)for everyq <∞.

Let us prove (18). By arguing as in the casem₊< m^∗=p, withmj=m₊, we arrive at n

i=1

Ω

min[u^am+, L^m+] · |∂_iu|^miλ

Ω

u^p·min[u^am+, L^m+]C, (19)

whereCis a positive constant independent ofLbecauseu∈L^am++p(Ω).

Assume thatL1, letΩ₁= {x∈Ω;u(x)1}and note that

min[u^am+, L^m+]min[u^ami, L^mi] a.e. inΩ\Ω1,∀i=1, . . . , n. (20)

Then, by (16), (19) and (20) we obtain (for constantsCindependent ofL) n

i=1

Ω\Ω1

∂_i

u·min[u^a, L]^miC n

i=1

Ω\Ω1

min[u^ami, L^mi] · |∂_iu|^miC. (21) On the other hand,

n i=1

Ω1

∂_i

u·min[u^a, L]^mi= n

i=1

(a+1)^mi

Ω1

u^ami· |∂_iu|^miC n i=1

Ω

|∂_iu|^mi<+∞. (22) Hence, combining (21) and (22) and lettingL→ ∞we obtain that

n i=1

Ω

∂i(u^a+1)^mi<+∞.

By Theorem 1,u∈L^(a+1)m∞(Ω), and (18) is proved.

Conclusion. Putf (x):=λu^p−1(x)so that alsof∈L^q(Ω)for allq <∞. Then, (1) reads −_n

i=1∂_i(|∂_iu|^mi−2∂_iu)=f inΩ

u=0 on∂Ω.

In view of [13, Theorem 2], we obtainu∈L^∞(Ω). 2

5. Proof of Theorems 3 and 4

By Theorem 2, for both statements it suffices to prove the existence of weak solutions.

We first prove Theorem 3. Letγ >0 and consider the minimization problem min

_n

i=1

Ω

|∂iu|^mi

mi ; u∈W₀^1,m(Ω),up=γ

. (23)

Consider a minimizing sequence{u_k}for (23). Since it is bounded inW₀^1,m(Ω), by Theorem 1 up to a subsequence {uk}converges inL^p(Ω)to someu. Clearly,up=γ, so thatu=0. By weak lower semicontinuity of the norm, we also infer

lim inf

k→∞ ∂_iu_kmi∂_iumi ∀i=1, . . . , n.

Therefore,uis a minimizer for (23) and there exists a Lagrange multiplierλ_γ >0 satisfying the requirements of the statement. Moreover,umay be taken nonnegative since|u|has the same norms asu. 2

The proof of Theorem 4 is obtained as a consequence of the mountain-pass Theorem [2] in its simplest form.

Therefore, we just quickly outline it.

On the spaceW₀^1,m(Ω)consider the functional J (u)=

n i=1

Ω

|∂iu|^mi mi −λ

p

Ω

|u|^p.

Theorem 1 ensures thatJ∈C¹(W₀^1,m(Ω)). Its Fréchet derivativeJ is defined by J (u), v

= n

i=1

Ω

|∂_iu|^mi−2∂_iu ∂_iv−λ

Ω

|u|^p−2uv ∀v∈W₀^1,m(Ω).

By elementary calculus, it is not difficult to show that there exists a constantC >0 independent ofsuch that ai>0∀i,

n i=1

ai=∈(0,1) ⇒ n i=1

a_i^mi

m_i C^m+. (24)

By the embeddingW₀^1,m(Ω)⊂L^p(Ω)we obtain J (u)

n i=1

Ω

|∂_iu|^mi

m_i −cu^p_1,m ∀u∈W₀^1,m(Ω).

This, combined with (24) by takinga_i= ∂_iumi, proves that there existsα, β >0 such that

J (u)α ∀u1,m=β. (25)

Moreover, ifu∈W₀^1,m(Ω)\ {0}andt >0 is sufficiently large, thenv:=tusatisfies

J (v) <0 and v1,m> β. (26)

Conditions (25) and (26) show that the functionalJ has a mountain-pass geometry.

Consider now a Palais–Smale sequence{u_k}forJ. It satisfies (for somec) J (u_k)→c and J (u_k)→0 in

W₀^1,m(Ω)

ask→ ∞. To derive a contradiction, assume thatu_k1,mdiverges; then,

o u_k1,m

=J (u_k)− 1 p

J (u_k), u_k

= n i=1

1 m_i − 1

p

Ω

|∂_iu_k|^mi

against the assumptionmi < pfor alli. Therefore, up to a subsequence,{uk}converges weakly inW₀^1,m(Ω)to someu. By the compact embedding stated in Theorem 1, we also have uk→u in L^p(Ω). Hence, since both J (uk), uk →0 andJ (uk), u →0, we have

n i=1

Ω

|∂iuk|^mi→λ

Ω

|u|^p= n i=1

Ω

|∂iu|^mi. (27)

By weak lower semicontinuity of the norms, for alliwe have lim inf_k∂_iu_kmi∂_iumi; this, together with (27) and weak convergence, shows thatuk→uinW₀^1,m(Ω). Therefore, the Palais–Smale condition holds.

As a straightforward consequence of the mountain-pass Theorem, we deduce thatJ admits a critical point.

SinceJ (u)=J (|u|)for allu, we may assume that such a critical point is nonnegative. This concludes the proof of Theorem 4. 2

6. Proof of Theorem 5

It is not restrictive to assume thatλ=1. The proof consists of four steps.

Step 1. There exists a unique functionw∈X:=W₀^1,m(Ω)∩L^p(Ω)which solves (8). For all test functions ϕ∈Xit satisfies

Ω

_n

i=1

|∂_iw|^mi−2+ε

1+ |Dw|²(m₋−2)/2

∂_iw

∂_iϕ+ |w|^p−2wϕ

=

Ω

f ϕ. (28)

In fact, Eq. (28) holds if and only ifJ (w)=0, whereJis the integral functional J (u)=

Ω

j (Du)+1

p|u|^p−f u

, u∈X,

withj (ξ ):=_n

i=1|ξi|^mi

mi +_m^ε₋(1+|ξ|²)^m−/2. If we endowXwith the weakW₀^1,m(Ω)topology, the functionalJis lower semicontinuous, becausejis convex and coercive. Thus, by the direct method of the calculus of variations,J admits at least one minimizerw∈X, which satisfies (28). Finally, the uniqueness is gained by the strict convexity of the functionalJ.

Step 2. The weak solutionwfound in Step 1 belongs toL^∞(Ω). Setk:=(sup|f|)^1/(p−1), andΩk:= {x∈ Ω: |w(x)|> k}. By takingϕ=(signw)max{|w| −k,0}as a test function in (28), we have:

Ω_k

n i=1

|∂iw|^mi

Ω_k

_n

i=1

|∂iw|^mi+ε

1+ |Dw|²(m₋−2)/2

|Dw|²

=

Ωk

|w| −k

f·(signw)− |w|^p−1

0,

which shows thatw∞k.

Step 3. The weak solutionwfound in Step 1 belongs toW_loc^1,∞(Ω)∩H_loc² (Ω). The first equation in (8) may be written in the form_n

i=1∂_ia_i(Du)=b(x), by setting a_i(ξ ):=

|ξ_i|^mi−2+ε

1+ |ξ|²(m₋−2)/2

ξ_i, b(x):= −f (x)+ |w|^p−2w. (29) The functionsai satisfy assumptions (2.3)–(2.6) of [18] (taking thereinp=m₋andq=m₊). The functionb(x) is inL^∞(Ω)(using the global boundedness ofwalready proved in Step 2). Finally, (4.2) in [18] is valid in view of (6) and (7). Hence, by Theorem 4.1 of [18], there exists a functionw∈W_loc^1,q(Ω)which satisfies, for everyΩ ⊂Ω,

Ω

_n

i=1

a_i(Dw)∂ _iϕ+b(x)ϕ

=0 ∀ϕ∈W₀^1,m+(Ω). (30)

We notice that alsowsatisfies (30); then, since the functionalJ is strictly convex, we deduce thatw=w. Using again Theorem 4.1 of [18], we deduce thatw∈W_loc^1,∞(Ω)∩H_loc² (Ω).

Step 4. The weak solutionwis of classC²(Ω).

The coefficientsai anda defined in (29) are differentiable in their variables, bounded with their first derivatives on every compact region ofΩ×R×Rⁿ, and satisfy a uniform ellipticity condition (

i,j∂_ja_i(ξ )η_iη_jε|η|²).

Therefore, the interiorC² regularity of w is obtained in a standard way, by applying the theory of uniformly elliptic equations (see e.g. [11, Chapters 13, 14, 15] or Theorems 6.2 and 6.3 in Chapter 4 of [15]). In order to obtain gradient bounds up to the boundary (which entailw∈C²(Ω)), one may either combine the interior gradient bound with the boundary Lipschitz estimate in [11, Theorem 14.1], or adapt the technique used by Lieberman in [16] or by Tolksdorf in [26]. 2

7. Proof of Theorem 6

The proof of Theorem 6 is inspired by a work of Otani [20], and is based on the construction of a sequence of

“doubly approximating” problems for (1). More precisely, letube a mild solution to (1). Letu_k =g_k(u), where theg_k(k∈N) areC¹(R⁺)functions such that

gk(s)=s ∀sk, gk(s)=k+1 ∀sk+2, 0g_k(s)1 ∀s0.